Regarding any religion being applied both politically and apolitically:
I can see Judaism, Christianity, and Islam as being in varying "stages of development" in the modern world.
That is to say, they vary in how far they have come, counting all of their followers, on the aggregate, in meeting the modern standard of separation between church and state.
Judaism, being the oldest, is the least political, then Christianity in the middle and Islam the youngest and most political.
I see that modern ideal as coming about from the Enlightenment, not instantly, but the Enlightenment making it possible and then the United States making it definite.
Even before that, there was an ostensible separation of religion and international relations (in Europe) due to the Treaty of Westphalia.
So it is more modern, and coming from the West which led the world into modernity, to have that greater separation of church and state, and I see the older Abrahamic religions as being further along in being "more mature" or "mellowed out".
Of course, followers of those religions would see it very differently, but from the outside looking in I'm making a fair generalization.
You're right to recognize the separation as only ever being ostensible too: how can a voter or elected official completely ignore their religious conscience in their political decisions? They can't.
We can remove blatant theocracy and keep out all religious language from laws, but law is formed through politics, and politics is informed by morality to some extent, and morality comes from religion for most people. At least, they personally view their morality as coming from their religion.
And right, as a Satanist I hold certain assumptions about human nature and justice and that influences how I think politically, as much as I try to linguistically separate my personal and my political theorizing.
And then another Satanist might also be influenced by his religion ultimately in his politics, but take totally opposite positions.
Regarding internal and external pressures messing with the status quo, that's definitely observable in the Muslim world, and has been especially since the end of WWI, when Western influence began to dramatically replace Ottoman influence in much of the Middle East.
The British and French bring new ideas of governance that clash with local religion and popular interpretation of Islam, but local Muslims want Western goods and that "corrupting influence" reduces the influence of Imams and tribal leaders, but if the governance of Muslim countries is seen as too secular the population resents it, and if it's too strictly Islamist (like Taliban style sharia) that's resented as well.
Then in Western countries, many Muslims clash with Christian and secular standards and Westerners resent that pressure.
Thank you for the reminder to keep a longer and more broad view of history, and not see the current problems of the society and time I live in as being particularly unique.
One of the reasons I enjoy these forums is to get this sort of dialogue with others and "thinking out loud" from myself to check my own thinking and conclusions.